A Hurwitz algebra or composition algebra is a finite-dimensional not necessarily associative algebra A with identity endowed with a nondegenerate quadratic form q such that q(a b) = q(a) q(b). If the underlying coefficient field is the reals and q is positive-definite, so that (a, b) = 1/2[q(a + b) − q(a) − q(b)] is an inner product, then A is called a Euclidean Hurwitz algebra or (finite-dimensional) normed division algebra.[4]
If A is a Euclidean Hurwitz algebra and a is in A, define the involution and right and left multiplication operators by
Evidently the involution has period two and preserves the inner product and norm. These operators have the following properties:
Similarly Re (ab)c = ((ab)c,1)1 = (ab, c*)1 = (b, a* c*)1 = (bc,a*)1 = (a(bc),1)1 = Re a(bc).
Hence ((ab)*, c) = (ab, c*) = (b, a*c*) = (1, b*(a*c*)) = (1, (b*a*)c*) = (b*a*, c), so that (ab)* = b*a*.
By the polarized identity ‖a‖2 (c, d) = (ac, ad) = (a* (ac), d) so L(a*) L(a) = L(‖a‖2). Applied to 1 this gives a*a = ‖a‖2 1. Replacing a by a* gives the other identity.
Substituting the formula for a* in L(a*) L(a) = L(a*a) gives L(a)2 = L(a2). The formula R(a2) = R(a)2 is proved analogously.
Classification
It is routine to check that the real numbers R, the complex numbers C and the quaternions H are examples of associative Euclidean Hurwitz algebras with their standard norms and involutions. There are moreover natural inclusions R ⊂ C ⊂ H.
Analysing such an inclusion leads to the Cayley–Dickson construction, formalized by A.A. Albert. Let A be a Euclidean Hurwitz algebra and B a proper unital subalgebra, so a Euclidean Hurwitz algebra in its own right. Pick a unit vectorj in A orthogonal to B. Since (j, 1) = 0, it follows that j* = −j and hence j2 = −1. Let C be subalgebra generated by B and j. It is unital and is again a Euclidean Hurwitz algebra. It satisfies the following Cayley–Dickson multiplication laws:
B and Bj are orthogonal, since j is orthogonal to B. If a is in B, then j a = a* j, since by orthogonal 0 = 2(j, a*) = ja − a*j. The formula for the involution follows. To show that B ⊕ B j is closed under multiplication Bj = jB. Since Bj is orthogonal to 1, (bj)* = −bj.
b(cj) = (cb) j since (b, j) = 0 so that, for x in A, (b(cj), x) = (b( jx), j(cj)) = −(b( jx), c*) = −(cb, ( jx)*) = −((cb) j, x*) = ((cb) j, x).
( jc)b = j(bc) taking adjoints above.
(bj)(cj) = −c*b since (b, cj) = 0, so that, for x in A, ((bj)(cj), x) = −((cj)x*, bj) = (bx*, (cj) j) = −(c*b, x).
Imposing the multiplicativity of the norm on C for a + bj and c + dj gives:
which leads to
Hence d(ac) = (da)c, so that Bmust be associative.
This analysis applies to the inclusion of R in C and C in H. Taking O = H ⊕ H with the product and inner product above gives a noncommutative nonassociative algebra generated by J = (0, 1). This recovers the usual definition of the octonions or Cayley numbers. If A is a Euclidean algebra, it must contain R. If it is strictly larger than R, the argument above shows that it contains C. If it is larger than C, it contains H. If it is larger still, it must contain O. But there the process must stop, because O is not associative. In fact H is not commutative and a(bj) = (ba) j ≠ (ab) j in O.[5]
Theorem. The only Euclidean Hurwitz algebras are the real numbers, the complex numbers, the quaternions and the octonions.
Other proofs
The proofs of Lee (1948) and Chevalley (1954) use Clifford algebras to show that the dimension N of A must be 1, 2, 4 or 8. In fact the operators L(a) with (a, 1) = 0 satisfy L(a)2 = −‖a‖2 and so form a real Clifford algebra. If a is a unit vector, then L(a) is skew-adjoint with square −I. So N must be either even or 1 (in which case A contains no unit vectors orthogonal to 1). The real Clifford algebra and its complexification act on the complexification of A, an N-dimensional complex space. If N is even, N − 1 is odd, so the Clifford algebra has exactly two complex irreducible representations of dimension 2N/2 − 1. So this power of 2 must divide N. It is easy to see that this implies N can only be 1, 2, 4 or 8.
This is a projective representation of a direct product of N − 1groups of order 2. (N is assumed to be greater than 1.) The operators Ui by construction are skew-symmetric and orthogonal. In fact Eckmann constructed operators of this type in a slightly different but equivalent way. It is in fact the method originally followed in Hurwitz (1923).[6] Assume that there is a composition law for two forms
where zi is bilinear in x and y. Thus
where the matrixT(x) = (aij) is linear in x. The relations above are equivalent to
Writing
the relations become
Now set Vi = (TN)tTi. Thus VN = I and the V1, ... , VN − 1 are skew-adjoint, orthogonal satisfying exactly the same relations as the Ui's:
Let G be the finite group generated by elements vi such that
where ε is central of order 2. The commutator subgroup[G, G] is just formed of 1 and ε. If N is odd this coincides with the center while if N is even the center has order 4 with extra elements γ = v1...vN − 1 and εγ. If g in G is not in the center its conjugacy class is exactly g and εg. Thus there are
2N − 1 + 1 conjugacy classes for N odd and 2N − 1 + 2 for N even. G has | G / [G, G] | = 2N − 1 1-dimensional complex representations. The total number of irreducible complex representations is the number of conjugacy classes. So since N is even, there are two further irreducible complex representations. Since the sum of the squares of the dimensions equals |G| and the dimensions divide |G|, the two irreducibles must have dimension 2(N − 2)/2. When N is even, there are two and their dimension must divide the order of the group, so is a power of two, so they must both have dimension 2(N − 2)/2. The space on which the Vi's act can be complexified. It will have complex dimension N. It breaks up into some of complex irreducible representations of G, all having dimension 2(N − 2)/2. In particular this dimension is ≤ N, so N is less than or equal to 8. If N = 6, the dimension is 4, which does not divide 6. So N can only be 1, 2, 4 or 8.
Applications to Jordan algebras
Let A be a Euclidean Hurwitz algebra and let Mn(A) be the algebra of n-by-n matrices over A. It is a unital nonassociative algebra with an involution given by
The trace Tr(X ) is defined as the sum of the diagonal elements of X and the real-valued trace by
TrR(X ) = Re Tr(X ). The real-valued trace satisfies:
These are immediate consequences of the known identities for n = 1.
It is trilinear and vanishes identically if A is associative. Since A is an alternative algebra[a, a, b] = 0 and [b, a, a] = 0. Polarizing it follows that the associator is antisymmetric in its three entries. Furthermore, if a, b or c lie in R then [a, b, c] = 0. These facts imply that M3(A) has certain commutation properties. In fact if X is a matrix in M3(A) with real entries on the diagonal then
with a in A. In fact if Y = [X, X 2], then
Since the diagonal entries of X are real, the off-diagonal entries of Y vanish. Each diagonal
entry of Y is a sum of two associators involving only off diagonal terms of X. Since the associators are invariant under cyclic permutations, the diagonal entries of Y are all equal.
Let Hn(A) be the space of self-adjoint elements in Mn(A) with product X ∘Y = 1/2(X Y + Y X) and inner product (X, Y ) = TrR(X Y ).
Theorem.Hn(A) is a Euclidean Jordan algebra if A is associative (the real numbers, complex numbers or quaternions) and n ≥ 3 or if A is nonassociative (the octonions) and n = 3.
To check that Hn(A) satisfies the axioms for a Euclidean Jordan algebra, the real trace defines a symmetric bilinear form with (X, X) = Σ ‖xij‖2. So it is an inner product. It satisfies the associativity property (Z∘X, Y ) = (X, Z∘Y ) because of the properties of the real trace. The main axiom to check is the Jordan condition for the operators L(X) defined by L(X)Y = X ∘Y:
This is easy to check when A is associative, since Mn(A) is an associative algebra so a Jordan algebra with X ∘Y = 1/2(X Y + Y X). When A = O and n = 3 a special argument is required, one of the shortest being due to Freudenthal (1951).[7]
In fact if T is in H3(O) with Tr T = 0, then
defines a skew-adjoint derivation of H3(O). Indeed,
so that
Polarizing yields:
Setting Z = 1 shows that D is skew-adjoint. The derivation property D(X ∘Y) = D(X)∘Y + X∘D(Y) follows by this and the associativity property of the inner product in the identity above.
With A and n as in the statement of the theorem, let K be the group of automorphisms of E = Hn(A) leaving invariant the inner product. It is a closed subgroup of O(E) so a compactLie group. Its Lie algebra consists of skew-adjoint derivations. Freudenthal (1951) showed that given X in E there is an automorphism k in K such that k(X) is a diagonal matrix. (By self-adjointness the diagonal entries will be real.) Freudenthal's diagonalization theorem immediately implies the Jordan condition, since Jordan products by real diagonal matrices commute on Mn(A) for any non-associative algebra A.
To prove the diagonalization theorem, take X in E. By compactness k can be chosen in K minimizing the sums of the squares of the norms of the off-diagonal terms of k(X ). Since K preserves the sums of all the squares, this is equivalent to maximizing the sums of the squares of the norms of the diagonal terms of k(X ). Replacing X by k X, it can be assumed that the maximum is attained at X. Since the symmetric groupSn, acting by permuting the coordinates, lies in K, if X is not diagonal, it can be supposed that x12 and its adjoint x21 are non-zero. Let T be the skew-adjoint matrix with (2, 1) entry a, (1, 2) entry −a* and 0 elsewhere and let D be the derivation ad T of E. Let kt = exp tD in K. Then only the first two diagonal entries in X(t) = ktX differ from those of X. The diagonal entries are real. The derivative of x11(t) at t = 0 is the (1, 1) coordinate of [T, X], i.e. a* x21 + x12a = 2(x21, a). This derivative is non-zero if a = x21. On the other hand, the group kt preserves the real-valued trace. Since it can only change x11 and x22, it preserves their sum. However, on the line x + y = constant, x2 + y2 has no local maximum (only a global minimum), a contradiction. Hence X must be diagonal.
Jacobson, N. (1968), Structure and representations of Jordan algebras, American Mathematical Society Colloquium Publications, vol. 39, American Mathematical Society
Jordan, P.; von Neumann, J.; Wigner, E. (1934), "On an algebraic generalization of the quantum mechanical formalism", Ann. of Math., 35 (1): 29–64, doi:10.2307/1968117, JSTOR1968117
Shapiro, Daniel B. (2000), Compositions of quadratic forms, De Gruyter Expositions in Mathematics, vol. 33, Walter de Gruyter, ISBN978-3-11-012629-7, Zbl0954.11011